To find the intervals where the company makes a profit or loses money, we need to analyze the given quadratic profit function:

$P(x) = -2x^2 + 402x - 4180$

Part 1: Finding when the company makes a profit

The company makes a profit when $P(x) > 0$. Since $P(x)$ is a quadratic function (a parabola that opens downward due to the negative coefficient of $x^2$), we can solve for the values of $x$ by finding where $P(x) = 0$, which gives the boundaries of the interval where $P(x) > 0$.

1. Set $P(x) = 0$ to find the roots: $-2x^2 + 402x - 4180 = 0$

2. Solve this quadratic equation using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = -2$, $b = 402$, and $c = -4180$.

   Substituting these values: $x = \frac{-402 \pm \sqrt{402^2 - 4(-2)(-4180)}}{2(-2)}$ $x = \frac{-402 \pm \sqrt{161604 - 33440}}{-4}$ $x = \frac{-402 \pm \sqrt{128164}}{-4}$ $x = \frac{-402 \pm 357.92}{-4}$

   Now, solve for both roots: $x_1 = \frac{-402 + 357.92}{-4} = \frac{-44.08}{-4} = 11.02 \approx 11$ $x_2 = \frac{-402 - 357.92}{-4} = \frac{-759.92}{-4} = 189.98 \approx 190$

So, the roots are $x_1 = 11$ and $x_2 = 190$.

Since the parabola opens downward, the profit is positive between these two roots. Therefore, the company makes a profit for:

$x \in (11, 190)$

Part 2: Finding when the company loses money

The company loses money when $P(x) < 0$. This happens outside the interval where the company makes a profit, meaning for values of $x$ less than 11 and greater than 190.

Thus, the company loses money for:

$x \in (-\infty, 11) \cup (190, \infty)$

Final Answer:

• Profit Interval: $(11, 190)$
• Loss Interval: $(-\infty, 11) \cup (190, \infty)$

Would you like further clarification on solving quadratic equations, or more details on the intervals?

Here are 5 related questions:

1. What are the general steps to solve a quadratic inequality like $P(x) > 0$?
2. How does the shape of a parabola determine where the function is positive or negative?
3. Can you explain how the discriminant helps determine the nature of the roots of a quadratic equation?
4. How do you use interval notation to describe the solution to inequalities?
5. What real-world applications might involve determining when a company makes a profit or loss?

Tip: Always check the direction the parabola opens (upward or downward) when interpreting the sign of the quadratic function in different intervals.